74.

ADDITION.

Addition in Algebra, is performed by connecting the Quantities by their proper Signs, and joining in any Sum fuch as can be united.

For performing which observe the following

RULE S.

1. If the Quantities to be added are alike, and have the fame Sign, add the Coefficients together, and to their Sum prefix the common Sign, and adjoin the common Letter or

Letters.

2. If the Quantities to be added are alike, but have unlike Signs, add together the Coefficients of the affirmative Terms (if there be more than one) and do the fame by the negative ones, and to their Difference prefix the Sign of the greater, adding the common Letter or Letters.

3. If the Quantities to be added are unlike, write them down after the other, with their proper Signs and Coefficients prefixed.

(1) To

EXAMPLE S.

6a+7b-3c (2) ab—6b+4x+109—152+6 Add 10db-7c 6ab b+x+ 49x+3

75.

SUBTRACTION.

Subtraction of Algebra is performed by the following general

RULE.

Change the Signs of the Quantity to be fubtracted into their contrary Signs, and then add it, fo changed, to the Quantity from which it was to be fubtracted (by the Rule of Addition) the Sum arifing will be the Remainder.

76. MULTIPLICATION. Multiplication of Algebra is also performed by the following general

RULE.

Multiply the Coefficients (if any) together, as in Sect. 4. and to their Product join the Letters, and prefix the proper Sign before them, which, when the Signs of the Factors are alike, that is, both, or both, the Sign of the Product is more; but when the Signs of the Factors are unlike, the Sign of the Product is

(7) Mul. 4x-5x+z (8) 2a-4b By -6f

(10) Mul. xx-ax By x+a

(9) aa+ab+bb a-b

2a+46

(11) xx+xy+xy xx-xy+xx

(13) √bc+dec

Vac

(15) 6cd√bad.

77.

DIVISION.

Divifion of Algebra Quantities is the direct Contrary to that of Multiplication, and confequently performed by direct contrary Operations.

RULE S.

1. When the Quantities in the Dividend have like Signs of thofe in the Divifor, and no Co-efficient in either, caft off all the Quantities in the Dividend, that are like thofe in the Divifor, and fet down the other Quantities with the Sign for the Quotient.

2. When the Quantities in the Dividend have unlike Signs to thofe in the Divifor, then fet down the Quotient Quantities found as in the laft Rule with the Sign

fore them.

be

3. If the Quantities in the Divifor cannot be exactly found in the Dividend, then fet them both down like a Vulgar Fraction, and find all the Quantities of the fame Letters that are in the Dividend and Divifor, and proceed with the Co-efficients, as in Cafe I. Sect. 36.

4. If the Quantity to be divided is compound, range its Parts according to the Dimensions of fome one of its Letters, and proceed as in Sect. V.

5. Different Powers or Roots of the fame Quantity are divided by fubtracting the Exponent of the Divifor from that of the Dividend, and place the Remainder as an Exponent to the Quantity given.

EXAMPLE S.

Divifor. Dividend.

(1.) d)ad-6d

(2.)-d)-ad-bd

(3.) a)aa+abl

(4.) -a)ab (5.) b)—ab-bd (6.) -bc) abcb c d + b c f ( (7.) 7b)42db (8.) 2bx)8abx-18bxc( (9.) 2b)ab+bb( (10.) 20a)10ab+15ac(

(11.) a—b)aaa-zaab3abb-bbbl

(12.) a+b)aa+2ab+bb( (13. a+b)aa—bb( (14.) 3a-6)6a^—96(

(15.) 3x2-4x+5)18x+-45x3+82x2—67x+40(

(16.) 4x-5a)48x376ax2-64a2x+105a3(

(17.) 3x+4a)81x1 —256aa (*

(18.) 2x-3a) 164x—72a2x2+-81a*(

(19.) 2xyz)4xy √xzz(

(20.) 20√2cyбoab √10acxy(

(21.) x2)x( (22.) a+x')a+x)'(

78.

FRACTION S.

Reduction of Algebraic Fractions are of the fame Nature, and require the fame Management as those of Num

bers.

A mixt Quantity is reduced to an improper Fraction by the Rules in Sect. 36, Cafe 3.

EXAMPLE S.

An improper Fraction is reduced to a mixt Quantity, by the Rule in Sect. 36, Cafe IV.

Fractions of different Denominations are reduced to Fractions of equal Value, and to have the fame Denominator by the Rule in Sect, 38, Cafe V.

Fractional Quantities are reduced into their lowest Terms by the Rule in Sect. 38, Cafe I.

EXAMPLE S.

The Rules for Addition, Subtraction, Multiplication, and Divifion of Algebraic Fractions, are the fame as for Numerical Fractions; fee Sect. 36, 37, 38, and 39.